Question

4. Fie a 0,5 +0,(3) +0,1(6) şi b=( 5/6-3/4)x12. Dintre următoarele relații, cea adevărată este:
a) a>1
b) b>=3/2
c)a=b
d) a + b < 2.

Answer 1

Răspuns:

c) a = b

Explicație pas cu pas:

a = 0,5 + 0,(3) + 0,1(6)

$a = \frac{5}{10} + \frac{3}{9} + \frac{15}{90} = \frac{5*9 + 3*10 + 15}{90} = \frac{90}{90} = 1$

$b = (\frac{5}{6} - \frac{3}{4} )*12$    aducem termenii din paranteză la același numitor: 12

$b = (\frac{5*2}{12} - \frac{3*3}{12} )*12 = \frac{10-9}{12} *12 = \frac{1}{12} *12 = 1$

Așadar, a = b = 1.

Dintre cele 4 relații, cea adevărată este c) a = b.

Celelalte relații sunt false.

Answer 2

$a=0,5+0,(3)+0,1(6);~~~b=\bigg(\dfrac{5}{6} -\dfrac{3}{4}\bigg)\times12\\\\\\a=0,5+0,(3)+0,1(6)= \dfrac{5}{10} +\dfrac{3}{9} +\dfrac{16-1}{90}=\dfrac{1}{2} +\dfrac{1}{3} +\dfrac{1}{6} =\\\\\\=\dfrac{3+2+1}{6}= \dfrac{6}{6} =1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{a=1}\\\\\\b=\bigg(\dfrac{5}{6}-\dfrac{3}{4}\bigg)\times12 =\dfrac{2\times5-3\times3}{12}\times12 =\dfrac{10-9}{12}\times12 =\\\\\\=\dfrac{1}{12}\times12=\dfrac{12}{12}=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{b=1}$

$a > 1\Rightarrow 1 > 1-Fals\\\\b\geq \dfrac{3}{2} \Rightarrow 1\geq 1,5-Fals\\\\\boxed{a=b\Rightarrow 1=1-Adevarat}\\\\a+b < 2\Rightarrow 1+1 < 2\Rightarrow 2 < 2-Fals\\\\Varianta~corecta~este~~c)~a=b.$